Questions tagged [line-bundles]

For questions about line bundles, that is vector bundles of rank $1$, over topological spaces.

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Exercise 4.4.2 from Huybrechts: Show $\int_X c_1(L)^n \ge 0$

The following is Huybrechts' Complex Geometry, Exercise 4.4.2: Show that for a base-point free line bundle $L$ on a compact complex manifold $X$ the integral $\int_X c_1(L)^n$ is non-negative. I've ...
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Divisors and line bundles

Let $X$ be a complex manifold and $D\subset X$ a smooth hypersurface, that is there exists an open covering $\{U_{\alpha}\}$ of $X$ and assign each $U_{\alpha}$ a holomorphic submersion $f_{\alpha}$, ...
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How to compute direct images of ramified covering maps?

Let $X$ be a smooth projective variety, $L$ a line bundle on $X$, and $s \in \Gamma(X,L) $ a non-zero section whose zero locus $D$ is smooth. Let $\pi: Y \to X$ be the $n$-sheeted ramified cover given ...
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Connections on indefinite inner product space bundles

Let $K$ denote a finite-dimensional vector space over $\mathbb{C}$. If $K$ is paired with a map $(\cdot,\cdot):K\times K\to\mathbb{C}$ such that, for all $\varphi,\psi,\chi\in K$ and $z,w\in\mathbb{C}$...
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Explicit section of explicit line bundle on elliptic curve

For a Riemann surface $\Sigma$ there is the following explicit construction of line bundles (from Donaldson: Riemann Surfaces, section 12.1.2): let $p \in \Sigma$. Let $D$ be a disk around $p$ with ...
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Admissible hermitian line bundles: two notions

In the framework of Arakelov geometry I came across two apparently different notions of admissible hermitian line bundle. I would like to understand why they are the same. Let $(X,\Omega)$ be a ...
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Evaluation Map of Line Bundle from Lazarsfeld I

In Chapter 1.1B, page 13, of Lazarsfeld's book Positivity in Algebraic Geometry I, he considers the following map. Let $L$ be a line bundle on a variety $X$ and $V\subset H^0(X, L)$ be a nonzero ...
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When is the anticanonical sheaf ample, on an arithmetic surface of genus 0?

Let $\pi:X \to S$ be an arithmetic surface, i.e. a flat, projective, scheme of relative dimension 1 over a Dedekind scheme $S$. So $X$ is a 2-dimensional excellent scheme. Suppose that the generic ...
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For an elliptic curve $X$, we have $X \cong \operatorname{Pic}^1(X)$. Why?

In Lectures on Vector Bundles, Le Potier states that for an elliptic curve $X$ we have $$ X \cong \operatorname{Pic}^1(X).$$ As far as I understand the map $x \mapsto [x]-[O]$ gives an isomorphism of $...
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Adjunction formula in Griffiths Harris

In the book Griffiths and Harris, they states that: $N^*_V=[-V]|_V$, here $V$ a smooth hypersurface, $N^*_V$ means the conormal bundle w.r.t. $V$. They define the transition function of $V$ as $g_{\...
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On global section of a line bundle.

At Proposition 1.1.1. (1) of https://arxiv.org/pdf/0706.0494.pdf it is written that $H^0(X, L \otimes {\cal{J}}(X, ||L||)) = H^0(X, L)$ ensures that every global holomorphic section $s$ of $L$, i.e., $...
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The first Stiefel-Whitney class of a trivial line bundle $E=M \times \mathbb{R}$

If the base space $M$ is non-orientable, is the trivial line bundle $E=M \times \mathbb{R}$ also non-orientable? i.e. $w_1(E) \neq 0$. If so, how could it be proved? Could we use $w_k(\xi\times\eta)=\...
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Line Bundles on Varieties

I am trying to construct line bundles on varieties. I've seen two definitions that work, but I can't see how to connect them. The first is what seems to be the classical definition of a variety $L$ ...
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Existence of a model for $(X,L)$

Recently I was reading Moriwaki's work about the heights on arithmetic varieties, and I came across the following doubt about the setting. Let $(X,L)$ be a couple where $X$ is a projective nonsingular ...
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Why do we have $\mu_* \mathcal{O}_Y(K_{Y/X}) = \mathcal{O}_X$ for a resolution?

Let $X$ be a normal complex variety and $\mu : Y \to X$ a resolution of singularities. We define the relative canonical divisor to be $K_{Y/X} := K_Y - \mu^* K_X$. In his book Positivity in Algebraic ...
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Projectivization of a bundle is invariant under tensoring with a line bundle

I want to prove that "Given a bundle $E$, for any line bundle $L$ the projectivizations of $E$ and $E$ tensor $L$ are isomorphic i.e $P(E)\cong P(E\otimes L)$". By bundle you can as well ...
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Pullback of tautological line bundle of $\Bbb{CP}^n$

Now let $\pi:\Bbb{C}^{n+1}\backslash\{0\}\to\Bbb{CP}^n$ be the natural projection, and $\mathcal{O}(-1)$ be its tautological line bundle, defined by $\{([x],l)\in\Bbb{CP}^n\times\Bbb{C}^{n+1}\mid l\in ...
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When the pull-back of positive line bundle by holomorphic map is also positive?

I'm finding condition under which the pullback of positive line bundle by holomorphic map is also positive. I'm reading the Daniel Huybrechts's Complex Geometry. In his book p.239, he define a ...
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Projective surface has an ample divisor

Let $S$ be a projective surface. In https://mathoverflow.net/questions/63999/nef-divisor-on-surface, there is an argument proving that any nef divisor $D$ on $S$ has $D^2\geq 0$. But the argument ...
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Type vs degree of a polarized abelian variety

Let $(A,L)$ be a polarized abelian variety. I know that the degree of the polarization is the Euler characteristic of $L$, so that $d = \chi(L) = \dim H^0(A,L)$ since $L$ is ample. I've read in a lot ...
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What is a non-effective holomorphic line bundle $L$ of degree 0 such that $L \otimes L$ is effective?

I am trying to solve an exercise that asks for a non-effective holomorphic line bundle $L$ of degree 0 such that $L \otimes L$ is effective. I think I have mostly solved it but since I am a bit shaky ...
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Twisted structure sheaves for $X=\operatorname{Spec} k$

Consider the twisted structure sheaves $\mathcal O_X(n), n\in \Bbb Z$ for X a closed subscheme of $\mathbb P^0_k$ for $k=\overline{k}$ a field. By definition, $\mathcal O_X(1):= i^*{\mathcal O_{\Bbb P^...
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Proof of the Theorem of the Cube

I am trying to understand the proof of the Theorem 2.5 here(Theorem of the Cube). It is required to prove that an invertible sheaf on $X\times Y\times Z$ is trivial when the restrictions to $\left \{ ...
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First Chern class coincides with degree of divisor without poincare duality or de rham cohomology

I know there are a lot of references (e.g. Griffiths-Harris page 141), but the issue is that these references always prove the proposition in arbitrary dimensions, using a somewhat contrived ...
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Some questions about an application of Grauert's theorem and the Picard scheme

The following is an exercise in Hartshorne. Let $ k $ be an algebraically closed field, $ Y $ an integral scheme of finite type over $ k $ and $ f : X \rightarrow Y $ a flat projective morphism whose ...
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Equivalence of two definitions for very ampleness

Hartshorne says an invertible sheaf $\mathcal{L}$ on $X$ is very ample relative to a field $k$ if there is an immersion $i:X\rightarrow \mathbb{P}_k^n$ for some $n$ such that $\mathcal{L}\simeq i^*\...
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Removing zero sections of n-twisted strips

First of all, a note: This question is motivated by my friend, who is a physics PhD student (I'm a math PhD student) and arose when I was trying to explain to him what line bundles are. (I wanted to ...
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1 answer
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Line subbundles of maximal degree

I want to know whether it is possible to bound the degree of line subbundles of certain holomorphic line bundles over Riemann surfaces. Even more concretely, consider the complex projective line $\...
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Picard group of blowup of smooth variety

If $X$ is a smooth irreducible variety and $Y$ is the blowup of $X$ at a point $p$ then Prove that $Pic(Y) = Pic(X)+$$\mathbb Z$ I was thinking about using the fact that the blowup map $p : Y \to X$ ...
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2 votes
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Line bundle associated to classifying map

I am trying to understand the correspondence between complex line bundles on $B$ and homotopy classes of maps $[B , BU(1)]$. One direction I can do. That is, if we have a map $f: B \to BU(1)$, we can ...
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Cohomology of pushforward of a line bundle

Let $X$ be a smooth projective surface and $C$ be a smooth curve on $X$ ( let $j$ denote the embedding of $C$ inside $X$). Let $L$ be a line bundle on $C$. Assume that it's known that $h^0( j_*(L) \...
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Why are at least three concurrent lines needed to specify a line bundle in 3D-Space?

I am doing visual SLAM and learning about Bundle Adjustment. So I started reading about bundles and pencils in geometry and I understood the main concept: In geometry, a pencil is a family of ...
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Global generation of a positive line bundle on a noncompact complex manifold

One says that a holomorphic line bundle $L$ is globally generated by the sections $s_{0}, \ldots, s_{N}$ if $\operatorname{Bs}\left(L, s_{0}, \ldots, s_{N}\right)=\emptyset$, i.e. those sections have ...
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Line bundle of point is twisting sheaf of Serre on projective line

I think I misunderstood something fundamental, but I cannot figure out where. Let $X = \mathbb{P}^1$ and let $D$ be the principal divisor $[0:1]$. I believe that $\mathcal{L}(D) = \mathcal{O}(1)$, ...
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  • 363
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Proving that a rational function is a section of an invertible sheaf

Let $S$ be a surface, $E$ be a curve on $S$, and $H$ be a hyperplane section of $S$. Let $a\in H^0(S,\mathcal O_S(H+(k-1)E))$, and $b\in H^0(S,\mathcal O_S(H+kE))$. Let $U$ be an open subset of $S$ on ...
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1 answer
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Intuitive concept of Levi-Civita connection.

For the line bundle $L$ over a manifold $M$, the connection $\nabla \colon L \to \Omega_M^1 \otimes L$ is defined. To the best of my knowledge when $L$ is a trivial line bundle ${\cal O}_M$, $\nabla$ ...
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2 votes
0 answers
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Dual of space of $\alpha$-densities on a vector space naturally isomorphism to $(-\alpha)$-densities

Let $V$ be an $n$-dimensional real vector space. For $\alpha \in \mathbb{R} \setminus \{0\}$, we define $\Omega^\alpha V$, the space of $\alpha$-densities on $V$, as the set of all maps $\omega : V^n \...
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When are calculations up to numerical equivalence appropriate?

William E. Lang writes in Examples of Surfaves of General Type with Vector Fields In the next two lemmas, we show that $K_X$ and $\mathcal O_X(D)$ are linear compinations of [some other bundles] . ...
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Which line bundle on $A$ is the pull-back of the Poincaré bundle via a given morphism $A \to A^*$?

Let $A$ be an Abelian variety (over an algebraically closed field of characteristic zero), and denote by $A^*$ its dual, which parameterizes degree zero line bundles on $A$. If $Q$ is a point of $A^*$,...
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How to think about the two quasi-coherent algebras $\bigoplus\limits_{n}\mathscr{E}^{\otimes n}, \,\, \bigoplus\limits_{n}\mathscr{E}(n)$?

Let $X$ be a scheme and $\mathscr{E}$ be a quasi-coherent $\mathcal{O}_X$-module. Take $X$ to be projective with a very ample bundle $\mathcal{O}_X(1)$ for a structure morphism $\pi: X \to S$, so that ...
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3 votes
1 answer
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The tensor product of the canonical line bundle and k(x) for a closed point x

I am reading the book “Fourier-Mukai transforms in algebraic geometry” by Daniel Huybrechts. At the beginning of the page 91, it is written that if $X$ is a smooth projective variety with a canonical ...
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Linear system of conics passing through $3$ points

I would like to understand why the set of classes of divisors corresponding to the conics of $\mathbb P^2$ passing through $3$ non-colinear points is a linear system of dimension $2$. Thank you very ...
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Why ample line bundle is positive? (and recommendation about understanding the Kodaira embedding theorem)

I'm beginner of Complex Geometry. Please Understand. I'm learning the Daniel Huybrechts's Complex Geometry. I want to understand the Kodaira Embedding Theorem as soon as possible since it provoke ...
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3 votes
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Construction of a complex surface through line bundles

In §2 of the paper [1] below, there is something I don't quite understand fully. Here, we take $F(x_0,x_1,x_2)$ to be homogeneous of degree $2k$ with real coefficients, with the additional assumption ...
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How to understand if we replace the fiber [-1,1] of a Möbius band by the complex number, the resulting bundle is trivial?

Trivial bundle means that the bundle is direct product of base manifold and the fibers, now it means the resulting bundle is $$E=S^{1}\times \mathbb{C}.$$ Some books only mentioned this result without ...
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Non trivial holomorphic section

Hello, let L be a holomorphic line bundle over a compact complex manifold of dimension 2. Suppose $\int_{X}c_{1}(L)^{2} > 0$ ($c_{1}$ means first Chern class). I would like to show $L^{\otimes m}$ ...
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1 vote
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Definition of the anticanonical bundle and references

I am looking for the definition of the anticanonical bundle of an algebraic variety. In the books, I only find the definition of the canonical bundle. Question 1 : Could anyone write me this ...
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Total Space of Trivial Line Bundle.

On Riemann sphere ${\Bbb P}_{\Bbb C}^1 = {\mathrm{Spec}}\, {\Bbb C}[X] \cup {\mathrm{Spec}}\, {\Bbb C}[Y]$, where the patching is given by $X \mapsto \frac{1}{Y}$, we can define the trivial line ...
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positive line bundle from topology viewpoint

We know that the existence of positive line bundle is a topological property(more explicitly, it's mertic free). When the line bundle is induced by a smooth hypersurface $V$ of the projective manifold ...
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Geometric Intuition for Ample Divisors and Ample Line Bundles

We know that a very ample divisor gives us an embedding into projective space and that very ample divisors are exactly the hyperplane sections. Is there a similar geometric intuition for ample ...
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